Lilith, Brick, Gaige and Krieg.
Lilith is just straight up overpowered, mathematically speaking. With the proper elemental loadout she can melt anything.
Brick is fun because you get to punch enemies so hard they explode.
Gaige... Screw the robot, her anarchy stacking system is too good. Max that shit out and you'll be blowing dudes away left and right, as and when you're able to actually hit them thanks to the accuracy loss.
Krieg is probably my favourite of the bunch because his mechanics encourage you to play like a crazy person. Yell at your friends to shoot you below a certain health threshold, then pop his super giant mutant form and tear everything up with a buzz axe rampage. Glorious.
(Also a neat touch is that his one-liners have a 'voice in your head' component that can only be heard client-side while playing with other people.)